Domination numbers and homotopy in certain ternary graphs

A ternary graph is a graph with no induced cycles of length $0$ modulo $3$. It was recently shown that, if the independence complex of a ternary graph is not contractible, then it is homotopy equivalent to a sphere. When a ternary graph also does not contain induced cycles of length $1$ modulo $3$, we prove that the dimension of the sphere is equal to the dimension of a minimum maximal simplex of the independence complex, or equivalently, to the value obtained by subtracting $1$ from the independent domination number of the graph. The same statement holds if we replace the independent domination number with the domination number. We also give a hypergraph analogue of the statement above.